Pythagoras Theorem

What is Pythagoras?

Pythagoras was a Greek mathematician who lived over 2500 years ago
He is most famous for Pythagoras’ Theorem, which includes the important formula for right-angled triangles

What is Pythagoras’ theorem?

Pythagoras’ theorem states that the longest side in a right-angled triangle is called the hypotenuse
- The hypotenuse will always be the side opposite the right angle
If we label the hypotenuse c, and label the other two sides a and b then Pythagoras’ Theorem tells us that:
- a² + b² = c²Where a, b, and c are the lengths of the three sides

Right Angled Triangle abc, IGCSE & GCSE Maths revision notes

How do I use Pythagoras’ theorem?

To find the length of the hypotenuse use $c equals square root of a squared plus b squared end root$
- This is just a rearrangement of the formula a ² + b ² = c ² , to make c the subject
- Note that when finding the hypotenuse you add inside the square root
To find the length of one of the other sides use $a equals square root of c squared minus b squared end root$ or $b equals square root of c squared minus a squared end root$
- This is a rearrangement of the formula a ² + b ² = c ² , to make either a or b the subject
- Note that when finding one of the shorter sides you subtract inside the square root

Exam Tip

‘Reality check’ your answers, if the hypotenuse ends up being shorter than another side in your answer, you’ve made a mistake somewhere
If your calculator gives you a ‘Math ERROR’ result, you probably subtracted the wrong way around in the square root
In questions with multiple steps, don’t round off (to 1 decimal place, 3 significant figures, etc) until the very end

Worked example

In the following diagram:
$A B space equals space 12 space cm$
$A C$ is a straight line, with $A D space equals space 9 space cm$ and $A C space equals space 22 space cm$

Find $x$ , the length of side $B C.$ Give your answer to 1 decimal place.

To find $x$ , we first need to find the length of $B D$ using triangle $A B D$
Note that $B D$ is a shorter side

$B D space equals space square root of 12 squared minus 9 squared end root space equals space square root of 63 space equals space 7.93725...$

It is best to leave rounding until the very end, use $square root of 63$ in subsequent working

Now we can find $x$ using triangle $B C D$

$D C space equals space 22 space minus space 9 space equals space 13 space cm$

$x equals square root of B D squared plus D C squared end root equals square root of open parentheses square root of 63 close parentheses squared plus 13 squared end root equals square root of 63 plus 169 end root$

$x equals square root of 232 space equals space 15.23154621...$

Round to 1 decimal place

$bold italic x bold equals bold 15 bold. bold 2 bold space bold cm$

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